Пусть 
0" alt="3^x=t\Rightarrow t>0" align="absmiddle" class="latex-formula">
![\frac{50t-100+\frac{50}{t}}{t+\frac{1}{t}+2}\leq\frac{15t-15}{t+1}+\frac{20+30t}{t+1}\\\frac{50t^2-100t+50}{t^2+2t+1}\leq\frac{45t+5}{t+1}\\\frac{50t^2-100t+50-(45t+5)(t+1)}{(t+1)^2}\leq0\\\frac{50t^2-100t+50-45t^2-50t-5}{(t+1)^2}\leq0\\\frac{5t^2-150t+45}{(t+1)^2}\leq0|*\frac{(t+1)^2}{5}\\t^2-30t+9\leq0\Rightarrow t\in[15-6\sqrt{6}; 15+6\sqrt{6}]](https://tex.z-dn.net/?f=%5Cfrac%7B50t-100%2B%5Cfrac%7B50%7D%7Bt%7D%7D%7Bt%2B%5Cfrac%7B1%7D%7Bt%7D%2B2%7D%5Cleq%5Cfrac%7B15t-15%7D%7Bt%2B1%7D%2B%5Cfrac%7B20%2B30t%7D%7Bt%2B1%7D%5C%5C%5Cfrac%7B50t%5E2-100t%2B50%7D%7Bt%5E2%2B2t%2B1%7D%5Cleq%5Cfrac%7B45t%2B5%7D%7Bt%2B1%7D%5C%5C%5Cfrac%7B50t%5E2-100t%2B50-%2845t%2B5%29%28t%2B1%29%7D%7B%28t%2B1%29%5E2%7D%5Cleq0%5C%5C%5Cfrac%7B50t%5E2-100t%2B50-45t%5E2-50t-5%7D%7B%28t%2B1%29%5E2%7D%5Cleq0%5C%5C%5Cfrac%7B5t%5E2-150t%2B45%7D%7B%28t%2B1%29%5E2%7D%5Cleq0%7C%2A%5Cfrac%7B%28t%2B1%29%5E2%7D%7B5%7D%5C%5Ct%5E2-30t%2B9%5Cleq0%5CRightarrow%20t%5Cin%5B15-6%5Csqrt%7B6%7D%3B%2015%2B6%5Csqrt%7B6%7D%5D "\frac{50t-100+\frac{50}{t}}{t+\frac{1}{t}+2}\leq\frac{15t-15}{t+1}+\frac{20+30t}{t+1}\\\frac{50t^2-100t+50}{t^2+2t+1}\leq\frac{45t+5}{t+1}\\\frac{50t^2-100t+50-(45t+5)(t+1)}{(t+1)^2}\leq0\\\frac{50t^2-100t+50-45t^2-50t-5}{(t+1)^2}\leq0\\\frac{5t^2-150t+45}{(t+1)^2}\leq0|*\frac{(t+1)^2}{5}\\t^2-30t+9\leq0\Rightarrow t\in[15-6\sqrt{6}; 15+6\sqrt{6}]")
Проверим левую границу на положительность:
0" alt="\sqrt{5,76}<\sqrt{6}<\sqrt{6,25}\Leftrightarrow 2,4<\sqrt{6}<2,5 \Leftrightarrow -15<-6\sqrt{6}<-14,4\Leftrightarrow\\\Leftrightarrow 0<15-6\sqrt{6}<0,6\Rightarrow 15-6\sqrt{6}>0" align="absmiddle" class="latex-formula">
Ответ: ![x\in[\log_{3}{(15-6\sqrt{6})}; \log_{3}{(15+6\sqrt{6})}]](https://tex.z-dn.net/?f=x%5Cin%5B%5Clog_%7B3%7D%7B%2815-6%5Csqrt%7B6%7D%29%7D%3B%20%5Clog_%7B3%7D%7B%2815%2B6%5Csqrt%7B6%7D%29%7D%5D "x\in[\log_{3}{(15-6\sqrt{6})}; \log_{3}{(15+6\sqrt{6})}]")